Transient Behavior of Single-Server Queuing Processes with Recurrent Input and Exponentially Distributed Service Times

Abstract

Customers arrive at a counter at the instants τ 1 , τ 2 , …, τ n …, where the inter-arrival times τ n − τ n −1 ( n = 1, 2, …, τ 0 = 0) are indentically distributed, independent, random variables. The customers will be served by a single server. The service times are identically distributed, independent, random variables with exponential distribution. Let ξ( t ) denote the queue size at the instant t . If ξ(τ n − 0) = k then a transition E k → E k +1 is said to occur at the instant t = τ n . The following probabilities are determined G n ( X ) = the probability that a busy period consists of n services and its length is at most x , and further the distribution of the number of transitions E k → E k +1 occurring in the time interval (0, t ).